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A tennis ball dipped in red paint rolls around on the coordinate plane, so that it is at

\((x,y) = (3t^2 - 9t - 5, t^2 - 3t + 2) \)
at time t, where \(0 \le t \le 4 \). Find the length of the paint trail left by the tennis ball.

 Jul 3, 2020
 #1
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As follows:

 

 Jul 3, 2020
 #2
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That's not correct is it Alan ? The function is multi (double ) valued.

The motion between t = 0 and t = 3/2 is retraced between t = 3/2 and t = 3,

and these will simply cancel out if the integral is taken between t = 0 and t = 4, (or between t = 0 and t = 3).

To see what's going on, you  have to integrate between 0 and 3/2, between 3/2 and 3 and between 3 and 4.

Guest Jul 3, 2020
 #3
avatar+30620 
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You are quite right.  I simply calculated the total distance travelled, without thinking carefully about the paint!

 

The graphs below show the various stages:

Alan  Jul 4, 2020

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